Enlargements in Geometry | 几何放大

📚 Enlargements in Geometry | 几何放大

Enlargement is a type of transformation that changes the size of a shape without altering its proportions. Unlike rotation, reflection or translation, an enlargement produces an image that is mathematically similar to the original object. The degree of enlargement is controlled by a scale factor, and the position of the image is determined by a fixed point known as the centre of enlargement. Mastering enlargements is essential for understanding similarity, map scales, and many real-world applications of geometry.

放大是一种改变图形大小但不改变其比例的变换。与旋转、反射或平移不同，放大产生的像是与原物体数学相似。放大的程度由比例因子控制，而像的位置由一个称为放大中心的固定点决定。掌握放大对于理解相似形、地图比例尺以及几何在现实中的许多应用至关重要。

In this revision guide, we will explore the key ideas of enlargement, starting from positive scale factors, moving through fractional factors that shrink a shape, and learning how to draw and describe enlargements accurately on a grid. We will use clear diagrams in words, structured examples, and practical tips that align with the Cambridge KS3 Mathematics curriculum.

在本复习指南中，我们将探索放大的关键概念，从正比例因子开始，到使图形缩小的分数因子，并学习如何在网格上准确地绘制和描述放大。我们将使用清晰的文字图解、结构化的示例和实用技巧，与剑桥KS3数学课程大纲保持一致。

1. What is an Enlargement? | 什么是放大？

An enlargement is a transformation that maps a shape to another shape with the same angles and proportional sides. The original shape is called the object, and the new shape is called the image. Every length in the image is multiplied by the same scale factor compared to the corresponding length in the object.

放大是将一个图形映射到另一个具有相同角度和比例边的图形的变换。原图形称为物体，新图形称为像。像中的每条边长都是物体中对应边长乘以相同的比例因子。

The transformation is different from translation because the image may change size and position, not just position. It differs from rotation and reflection because those preserve size, while enlargement changes size unless the scale factor is 1. The shape remains similar to the original, which is why enlargement is sometimes called a similarity transformation.

这种变换与平移不同，因为像可能改变大小和位置，而不仅仅是位置。它与旋转和反射的区别在于后者保持大小不变，而放大会改变大小（除非比例因子为1）。图形保持与原图相似，这就是放大有时被称为相似变换的原因。

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2. Scale Factor | 比例因子

The scale factor, often denoted by k, tells you how many times larger or smaller the image is compared to the object. If k > 1, the image is larger than the object; if 0 < k < 1, the image is smaller; if k = 1, the image is congruent to the object (no size change). Negative scale factors produce an inverted image, but in KS3 we typically focus on positive scale factors.

比例因子，通常用k表示，告诉你像相对于物体放大了或缩小了多少倍。如果k > 1，像比物体大；如果0 < k < 1，像比物体小；如果k = 1，像与物体全等（大小不变）。负比例因子会产生倒置的像，但在KS3阶段我们通常只关注正比例因子。

To find the scale factor when you are given the object and the image, pick a side length from the image and divide it by the corresponding side length on the object. For example, if a side of the object is 3 cm and the corresponding side of the image is 9 cm, the scale factor is 9 ÷ 3 = 3. Always check that all lengths are multiplied by the same scale factor; if not, the shapes are not similar.

当给出物体和像时，求比例因子：从像中选一条边长除以物体上对应的边长。例如，物体的一条边为3厘米，像的对应边为9厘米，则比例因子为9 ÷ 3 = 3。务必检查所有长度是否都乘以相同的比例因子；如果不是，这两个图形不相似。

image length = object length × k

像的长度 = 物体的长度 × k

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3. Centre of Enlargement | 放大中心

The centre of enlargement is the fixed point from which rays are drawn through the vertices of the object to determine the position of the image. The centre can be inside, on the edge, or outside the shape. When you enlarge a shape, each vertex of the image lies on the ray that starts at the centre and passes through the corresponding vertex of the object, and the distance from the centre is scaled by k.

放大中心是固定的点，从其出发经过物体顶点的射线用来确定像的位置。中心可以在图形内部、边上或外部。当你放大图形时，像的每个顶点都位于以中心为起点、经过物体对应顶点的射线上，且到中心的距离按比例因子k缩放。

For example, if the centre of enlargement is at (0,0) and a vertex of the object is at (2,3) with scale factor 2, the image vertex will be at (2×2, 3×2) = (4,6). Thus, the centre acts as an anchor that controls both the size and the position of the image.

例如，如果放大中心在(0,0)，物体的一个顶点在(2,3)，比例因子为2，则像的顶点位于(2×2, 3×2) = (4,6)。因此，中心起着锚点的作用，控制着像的大小和位置。

To find the centre when given the object and image, draw straight lines connecting corresponding vertices. All these lines will intersect at a single point – that intersection is the centre of enlargement. This is a useful skill for describing transformations completely.

当给出物体和像求中心时，连接对应顶点画直线。所有这些直线会相交于一点——这个交点就是放大中心。这是完整描述变换的实用技能。

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4. Positive Scale Factors Greater Than 1 | 大于1的正比例因子

When the scale factor is greater than 1, the image is larger than the object. For example, scale factor 2 doubles every side length, scale factor 3 triples them, and so on. The shape is stretched away from the centre of enlargement, but the angles stay identical.

当比例因子大于1时，像比物体大。例如，比例因子2使每条边长加倍，比例因子3使其变为三倍，以此类推。图形远离放大中心拉伸，但角度保持不变。

Imagine a triangle with vertices A(1,2), B(3,2), C(2,4) and enlarge it by scale factor 2 from the centre (0,0). Each coordinate is multiplied by 2: A'(2,4), B'(6,4), C'(4,8). The new triangle is twice as far from the origin in every direction, and its area becomes 4 times larger because area scales by k².

想象一个三角形顶点为A(1,2)、B(3,2)、C(2,4)，从中心(0,0)以比例因子2进行放大。每个坐标都乘以2：A'(2,4)、B'(6,4)、C'(4,8)。新三角形在每个方向上到原点的距离都变为原来的两倍，而其面积变为原来的4倍，因为面积按k²缩放。

Always remember to multiply both the x- and y-coordinates separately when working on a coordinate grid. Keeping a ray diagram in mind helps avoid confusion about which direction the image should move.

在坐标网格上操作时，务必记得分别将x坐标和y坐标相乘。在脑海中保持射线图有助于避免对像应往哪个方向移动产生混淆。

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5. Scale Factor Between 0 and 1 (Fractional Enlargement) | 0和1之间的比例因子（分数放大）

A scale factor that is a fraction between 0 and 1 shrinks the shape, making the image smaller than the object. This is sometimes called a ‘reduction’ or ‘fractional enlargement’. For instance, a scale factor of ½ halves all side lengths, producing an image that is half the size of the original.

介于0和1之间的分数比例因子会缩小图形，使像比物体小。这有时被称为“缩小”或“分数放大”。例如，比例因子½会将所有边长减半，生成一个大小为原图一半的像。

Consider an object rectangle with length 8 cm and width 4 cm. Enlarging by scale factor ⅓ gives an image with length 8 × ⅓ = 2⅔ cm and width 4 × ⅓ = 1⅓ cm. Notice that the centre of enlargement still lies on the lines joining corresponding vertices, but the image appears closer to the centre than the object.

考虑一个物体矩形，长8厘米，宽4厘米。以比例因子⅓放大，得到像的长为8 × ⅓ = 2⅔厘米，宽为4 × ⅓ = 1⅓厘米。注意放大中心仍在对应顶点连线上，但像看起来比物体更靠近中心。

On a coordinate grid, if the centre is (0,0) and a point is at (9,6), applying a scale factor of ⅓ moves it to (9×⅓, 6×⅓) = (3,2). The image is still similar, with unchanged angles, just smaller.

在坐标网格上，如果中心是(0,0)，某点在(9,6)，应用比例因子⅓会将其移到(9×⅓, 6×⅓) = (3,2)。像仍然是相似的，角度不变，只是更小。

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6. Drawing an Enlargement Step by Step | 逐步绘制放大图形

To draw an enlargement accurately, follow these steps: (1) mark the centre of enlargement C; (2) draw a ray from C through a vertex of the object; (3) measure the distance from C to the vertex; (4) multiply this distance by the scale factor k; (5) mark the new vertex at that distance along the same ray; (6) repeat for all vertices; (7) join the image vertices in the correct order.

要精确地绘制放大图形，请按以下步骤操作：(1) 标出放大中心C；(2) 从C经过物体的一个顶点画一条射线；(3) 测量从C到该顶点的距离；(4) 将该距离乘以比例因子k；(5) 在同一条射线上距离C为计算结果的位置标出新顶点；(6) 对所有顶点重复；(7) 按正确顺序连接像的顶点。

If the centre of enlargement is on the shape, some vertices might remain fixed (if the distance becomes zero by coincidence), but generally, all move along rays. When working on squared paper, you can count squares horizontally and vertically from the centre, then multiply those counts by k to locate the image vertices.

如果放大中心在图形上，有些顶点可能保持不变（如果巧合下距离变为零），但一般来说所有顶点都会沿射线移动。在方格纸上操作时，可以从中心开始水平和垂直地数格子，然后将这些格数乘以k来定位像的顶点。

A common mistake is to forget to use the centre and instead just multiply side lengths without adjusting position. This changes size correctly but gives the wrong location, making the transformation incomplete.

一个常见错误是忘记使用放大中心，而仅仅将边长相乘却不调整位置。这样做虽然尺寸变化正确，但位置错误，导致变换不完整。

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7. Finding the Scale Factor and Centre from an Object and Image | 从物体和像求比例因子和中心

When you are given a drawn object and its image, you may be asked to describe the enlargement fully. The scale factor k is found by dividing a length on the image by the corresponding length on the object. Confirm this ratio is the same for at least one other pair of corresponding lengths to be sure.

当你看到给出的物体及其像时，可能需要完整描述该放大。比例因子k通过将像上的一条边长除以物体上对应边长来求得。确认至少另一对对应边长也具有相同的比值以确保准确。

To find the centre of enlargement, draw straight lines through at least two pairs of corresponding vertices. The point where these lines intersect is the centre. The centre can lie anywhere – inside the shapes, between them, or even outside. Label it clearly as the centre.

要求放大中心，则至少通过两对对应顶点画直线。这些直线的交点就是中心。中心可以位于任何位置——在图形内部、它们之间，甚至外部。清晰地将其标注为中心。

In rare cases, the object and image might be the same size but in different positions; that suggests a scale factor of 1 and the centre is infinitely far away, which is actually a translation, not an enlargement. So it is important to check the scale factor first.

在极少数情况下，物体和像大小相同但位置不同，这暗示比例因子为1而中心无限远，实际上这是平移而非放大。因此首先检查比例因子非常重要。

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8. Enlargements on a Coordinate Grid | 坐标网格上的放大

Enlargements on a coordinate grid are very common in KS3. When the centre of enlargement is the origin (0,0), simply multiply each coordinate of the object by the scale factor: (x, y) → (kx, ky). This method works for both enlargement and reduction.

坐标网格上的放大在KS3中非常常见。当放大中心为原点(0,0)时，只需将物体的每个坐标乘以比例因子：(x, y) → (kx, ky)。这种方法对放大和缩小都适用。

If the centre is not the origin, you must find the vector from the centre to the vertex, multiply that vector by k, and add the result to the centre’s coordinates. For example, with centre (2,1) and vertex (5,3), the vector is (5-2, 3-1) = (3,2). Multiply by k=2 → (6,4), then add to centre: (2+6, 1+4) = (8,5).

如果中心不是原点，你必须求出从中心到顶点的向量，将该向量乘以k，然后将结果加到中心的坐标上。例如，中心(2,1)和顶点(5,3)，向量为(5-2, 3-1) = (3,2)。乘以k=2 → (6,4)，再加到中心：(2+6, 1+4) = (8,5)。

Practising coordinate enlargements helps students link algebraic ideas with geometric transformations. It also prepares them for the concept of similar triangles and future work on vectors.

练习坐标放大有助于学生将代数思想与几何变换联系起来，也为他们学习相似三角形的概念和未来的向量内容做好准备。

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9. Relationship Between Enlargement and Similarity | 放大与相似的关系

When two shapes are similar, one is an enlargement of the other. This means corresponding angles are equal and corresponding sides are in the same ratio. The scale factor of enlargement is that constant ratio. If triangle ABC is an enlargement of triangle DEF with scale factor 1.5, then AB/DE = BC/EF = CA/FD = 1.5.

当两个图形相似时，一个是另一个的放大。这意味着对应角相等，对应边成相同比例。放大比例因子就是那个恒定的比值。如果三角形ABC是三角形DEF的放大且比例因子为1.5，则AB/DE = BC/EF = CA/FD = 1.5。

This connection is powerful: you can use enlargement to solve problems in similarity, such as finding missing lengths in similar figures. If a scale model of a building is made with a scale factor of 1/100, every real length is 100 times the model length. The shape, however, remains the same.

这种联系非常有用：你可以利用放大来解决相似问题，例如求相似图形中的缺失长度。如果一座建筑的比例模型以1/100的比例因子制作，那么每个实际长度都是模型长度的100倍。但形状保持不变。

Understanding enlargement therefore strengthens your grasp of proportional reasoning and prepares you for later topics like trigonometry, where similar triangles play a central role.

因此，理解放大能增强你对比例推理的掌握，并为后续的主题如三角学做好准备，其中相似三角形起着核心作用。

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10. Calculating Area and Volume After Enlargement | 计算放大后的面积和体积

When a 2D shape is enlarged by a scale factor k, the area of the image becomes k² times the area of the object. For example, if a square of side 2 cm (area 4 cm²) is enlarged by scale factor 3, the new square has side 6 cm and area 36 cm², which is 9 (3²) times the original area.

当一个二维图形以比例因子k放大时，像的面积变为物体面积的k²倍。例如，一个边长为2厘米的正方形（面积4平方厘米）以比例因子3放大，新正方形的边长为6厘米，面积为36平方厘米，这是原面积的9（3²）倍。

Similarly, for 3D shapes, volume scales by k³. If a cube of side 1 cm is enlarged by factor 2, its volume changes from 1 cm³ to 8 cm³ (2³ = 8). This principle is essential in scale models and in understanding how surface area and volume change in real-world enlargements.

类似地，对于三维图形，体积按k³缩放。如果一个边长为1厘米的立方体以比例因子2放大，其体积从1立方厘米变为8立方厘米（2³ = 8）。这一原理在比例模型以及理解现实放大中表面积和体积如何变化时至关重要。

What scales?	Factor	Example (k=2)
Length	k	2 times
Area	k²	4 times
Volume	k³	8 times

缩放量 | 因子 | 示例(k=2)
长度 | k | 2倍
面积 | k² | 4倍
体积 | k³ | 8倍

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11. Common Mistakes and How to Avoid Them | 常见错误及如何避免

One frequent error is confusing enlargement with stretching in only one direction; true enlargement changes all dimensions proportionally. Another is forgetting to use the centre of enlargement when describing the transformation fully – simply stating ‘it’s enlarged by scale factor 2’ is incomplete without the centre.

一个常见错误是混淆放大与仅在一个方向上拉伸；真正的放大是按比例改变所有尺寸。另一个错误是在完整描述变换时忘记给出放大中心——只说“以比例因子2放大”而不提及中心是不完整的。

Students sometimes calculate the scale factor by dividing object length by image length instead of the other way round. Remember: image length divided by object length gives k. Also, when drawing, failing to project rays accurately from the centre leads to a distorted image. Use a sharp pencil and a ruler, and double-check distances.

学生有时计算比例因子时用物体边长除以像边长，而不是反过来。记住：像边长除以物体边长得到k。另外，在绘制时，未能准确地从中心投射射线会导致像失真。使用尖铅笔和直尺，并复查距离。

Finally, when finding the centre, do not rely on only one pair of vertices – use at least two to confirm the intersection point. If the lines appear to meet at a single point, you have correctly identified the centre.

最后，在寻找中心时，不要只依赖一对顶点——至少使用两对以确认交点。如果直线看起来汇聚于一点，你就正确地找到了中心。

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12. Practice Questions and Real-Life Applications | 练习题及实际应用

Enlargements appear everywhere: map scales, photocopiers enlarging documents (e.g., A4 to A3), architectural scale drawings, and even digital zoom on a camera. When a map uses a scale of 1 : 25,000, the real distance is an enlargement of the map distance with scale factor 25,000.

放大无处不在：地图比例尺、复印机放大文件（例如A4放大到A3）、建筑比例图，甚至相机的数码变焦。当一张地图使用1:25,000的比例时，实际距离是地图距离的放大，比例因子为25,000。

In biology, the image from a microscope is an enlargement of the specimen. If the magnification is 400×, the apparent distance is multiplied by 400. Understanding enlargement helps scientists measure real lengths from photographs.

在生物学中，显微镜下的图像就是标本的放大。如果放大倍数为400×，表观距离被乘以400。理解放大有助于科学家从照片中测量实际长度。

Try these quick checks: (1) Enlarge point P(3, -2) by scale factor 4 with centre (0,0). Answer: P'(12, -8). (2) A rectangle 5 cm by 3 cm is reduced by scale factor 0.6; find the new dimensions. Answer: 3 cm by 1.8 cm. (3) Two similar triangles have corresponding sides 7 cm and 21 cm; what is the scale factor? Answer: 21 ÷ 7 = 3.

试试这些快速自测题：(1) 将点P(3, -2)以中心(0,0)按比例因子4放大。答案：P'(12, -8)。 (2) 一个5厘米×3厘米的矩形以比例因子0.6缩小；求新尺寸。答案：3厘米×1.8厘米。 (3) 两个相似三角形对应边长为7厘米和21厘米；比例因子是多少？答案：21 ÷ 7 = 3。

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